Page 40 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

1.2 brieF history oF the study oF Vibration 37
It has long been recognized that many basic problems of mechanics, including those
of vibrations, are nonlinear. Although the linear treatments commonly adopted are quite
satisfactory for most purposes, they are not adequate in all cases. In nonlinear systems,
phenomena may occur that are theoretically impossible in linear systems. The mathemati-
cal theory of nonlinear vibrations began to develop in the works of Poincaré and Lyapunov
at the end of the nineteenth century. Poincaré developed the perturbation method in 1892
in connection with the approximate solution of nonlinear celestial mechanics problems. In
1892, Lyapunov laid the foundations of modern stability theory, which is applicable to all
types of dynamical systems. After 1920, the studies undertaken by Duffing and van der Pol
brought the first definite solutions into the theory of nonlinear vibrations and drew atten-
tion to its importance in engineering. In the last 40 years, authors like Minorsky and Stoker
have endeavored to collect in monographs the main results concerning nonlinear vibra-
tions. Most practical applications of nonlinear vibration involved the use of some type of a
perturbation-theory approach. The modern methods of perturbation theory were surveyed
by Nayfeh [1.11].
Random characteristics are present in diverse phenomena such as earthquakes, winds,
transportation of goods on wheeled vehicles, and rocket and jet engine noise. It became
necessary to devise concepts and methods of vibration analysis for these random effects.
Although Einstein considered Brownian movement, a particular type of random vibration,
as long ago as 1905, no applications were investigated until 1930. The introduction of the
correlation function by Taylor in 1920 and of the spectral density by Wiener and Khinchin
in the early 1930s opened new prospects for progress in the theory of random vibrations.
Papers by Lin and Rice, published between 1943 and 1945, paved the way for the applica-
tion of random vibrations to practical engineering problems. The monographs of Crandall
and Mark and of Robson systematized the existing knowledge in the theory of random
vibrations [1.12, 1.13].
Until about 40 years ago, vibration studies, even those dealing with complex
engineering systems, were done by using gross models, with only a few degrees of
freedom. However, the advent of high-speed digital computers in the 1950s made
it possible to treat moderately complex systems and to generate approximate solu-
tions in semidefinite form, relying on classical solution methods but using numerical
evaluation of certain terms that cannot be expressed in closed form. The simultaneous
development of the finite element method enabled engineers to use digital computers
to conduct numerically detailed vibration analysis of complex mechanical, vehicu-
lar, and structural systems displaying thousands of degrees of freedom [1.14, 1.16].
Although the finite element method was not so named until recently, the concept was
used centuries ago. For example, ancient mathematicians found the circumference of
a circle by approximating it as a polygon, where each side of the polygon, in present-
day notation, can be called a finite element. The finite element method as known
today was presented by Turner, Clough, Martin, and Topp in connection with the
analysis of aircraft structures [1.15]. Figure 1.5 shows the finite element idealization
of a milling machine structure.